# AIC for increasing sample size

I am using AIC as a model selection criteria in one of my projects. However, since AIC isn't dependent on the number of points sampled, for large n the log likelihood term rapidly outscales the parameter penalty.

I was wondering why the parameter penalty doesn't scale with the number of points, as the log likelihood generally does. It's getting to where the log likelihood is in the order of tens of thousands and the AIC penalty for having ~10 extra parameters in the model doesn't matter. But it feels like it really should. Am I misunderstanding something?

• Why would having 10 extra parameters matter if you have enough data to estimate them rather precisely? AIC/n (AIC per datapoint) estimates the log-likelihood of a new data point from the same population; when you have enough data, this is approximately equal to the average sample likelihood (log-likelihood/n) as the estimation error for the parameters is negligible. Apr 5 '19 at 13:19
• Sorry, I don't think I articulated my question very well. Let's say you have many points of somewhat noisy data. Adding a decent number of parameters (lets stay 10) to your model will likely be very beneficial to your log likelihood. However, the -2k part of the AIC calculation will barely penalize the model for it. It just seems to me that the AIC doesn't appropriately penalize for extra params. Apr 5 '19 at 13:51
• In my comment above, it should be negative likelihood, not raw likelihood. Apr 5 '19 at 15:22

## 1 Answer

It's a known criticism of AIC.

The BIC scales the penalty of number of model parameters by the root of n. In even larger sample sizes,

$$\text{BIC} = \log(n) k - 2 \log \mathcal{L},$$

though you will still tend to find BIC favors models with more parameters in larger samples. In either case, it's a desirable trait of model selection criteria that tends to select more parameters in larger sample sizes. It all boils down to how many you want to enter into a particular model for a particular sample size. When that's a finite number, there's no reason to use information criteria at all.

Shibata's work on AIC works under the concept of "mean efficiency". That is: ICs work under the condition that you know or assume that the number of variables in an ideal model is infinitely valued, and that in larger samples you will tend to favor models with more variables.

• You can criticize a hammer if your problem does not look like a nail, but I wonder if there is any ground for criticizing the design of AIC taking into account what it actually aims for. After all, AIC is an efficient model selection criterion, which BIC and other criteria with relatively fast increasing penalties are not. So if your goal is optimal prediction (optimal in terms of maximizing the likelihood of a new observation), AIC will do it for you. If your goal is not prediction, why would you be considering AIC to begin with? Does that make sense? Apr 5 '19 at 15:19
• OK, I guess you can justify your criticism of the assumption of infinitely many parameters in the "ideal model", as you mention in your last paragraph. So then the question would be, does my problem look like one where this assumption may hold or not? If so, AIC is fine, if not, go look for another information criterion. Apr 5 '19 at 15:26
• @RichardHardy We agree on all points. The revelation that AIC only works in some very contrived situations won't stop people from asking whether it functions well in other situations. The answer, aside from "it wasn't meant to do that" is "it doesn't do that very well". It's a revelation that another inappropriate tool (BIC) "does it a bit better". There are much, much better tools for data reduction if OP wants a "sparse number of predictors in a reasonably large sample", but it wasn't the question that was asked. Apr 5 '19 at 15:41
• Good. I would contest, however, your use of "very contrived situations", or even "contrived situations". A large (the largest?) part of real world phenomena are results of infinitely complex data generating processes which require an infinite amount of parameters to be fully charaterized, which is exactly what the premise of AIC is. Hence, as long as the goal is optimal prediction, AIC strikes me as the most reasonable choice, or at least a solid baseline. When prediction is not the goal while, say, finding a sparse number of predictors is, we need other tools. Apr 5 '19 at 16:35